So next, I would like to describe the action of the Steenrod squares in a more general context (which is pure homological algebra). This more general approach to Steenrod operations seems to be due to Peter May, but I learned it from these lecture notes by Jacob Lurie.

As I said earlier, the Steenrod squares will act on the cohomology of any -algebra. What is an -algebra? It’s supposed to be a dga with a multiplication law that is coherent and associative up to coherent homotopy. Or, equivalently, an algebra over an -operad in chain complexes. Unfortunately, I don’t know too many simple ones. For instance, the standard -operad in *spaces*, the infinite little cubes operad, is huge: the individual terms have to be contractible spaces large enough to admit a free action of the symmetric group.

But we’ll need less structure than an -algebra structure. We’ll need a multiplication law on a chain complex which is commutative up to *coherent homotopy*, but without quite all the coherence that one would need for an -algebra.

What should this be? Well, a multiplication law would be a map , and to say that it is commutative would be to say that it is -equivariant where acts by permutation on and trivially on . In other words, it would be a map where denotes the coinvariants. This is generally going to be too strong: most of the homotopy commutative multiplication laws we deal with will not be rectifiable to strictly commutative ones.

Instead, we’ll use maps not out of the coinvariants of , but out of the *homotopy coinvariants.* To get this, one first tensors with a complex of free -modules which is contractible as a complex of vector spaces, and then takes the coinvariants.

More generally:

Definition 1If is a -equivariant complex of -vector spaces, then we define thehomotopy coinvariantsas the coinvariants of . Here is any acyclic resolution of the complex (just in degree zero, nothing elsewhere) consisting of free -modules.